Allocating the Cost of Capital

1. Allocating the Cost of Capital CAS Spring Meeting May 19-22, 2002 Robert P. Butsic Fireman’s Fund Insurance

2. Why is Capital Necessary? • The answer is not obvious: • We can’t have enough capital to eliminate insurance insolvency • So why not have minimal capital and let guaranty funds protect policyholders? • Answer is: frictional insolvency costs • Additional system costs due to insolvency: • Legal fees, market disruption, extra claims handling costs

3. Optimal Capital Level Frictional Insolvency Costs Frictional Capital Costs Capital Amount

4. What is the Cost of Capital? • Investor supplies capital and expects return commensurate with risk to which the capital is exposed • This return is the cost of capital • Traditional view of insurance management • But, look from the modern finance perspective

5. Base Cost of Capital • A: Investor invests capital in a levered fund • borrow cash and invest all assets • identical to the insurer’s assets • Investor’s expected return in A is called Base Cost of Capital • B: Insurer has same balance sheet • But insurer has higher COC

6. Frictional Costs of Capital • The insurance mechanism will introduce extra costs • Government, regulation and organization • Illiquid nature of insurance liability • Information asymmetry (opaqueness) • These are frictional costs of capital • key one is double taxation • Most easily quantified

7. Double Taxation Example • Investor can directly invest in security with 10% return, but invests in ABC Insurance, who puts money in same security • ABC gets 10% return, pays 35% tax and gives 6.5% net back to investor • A losing deal unless PH can make up the difference

8. Other Frictional Costs • Regulatory costs • Capital can’t be easily moved, so investment is illiquid • Agency costs • Misalignment of owners’ and managers’ interests (Enron a classic example) • Financial distress costs • Legal fees • Distressed sale of assets

9. Financial Pricing Model • Fair premium = total present value of • loss & LAE (including risk margin) • UW expenses • Frictional capital costs • Note that traditional (base) cost of capital is embedded in risk margin

10. Risk Margin and COC Example • Assumptions • Fair premium is $1000, paid up front, $1040 loss paid in one year • Risk-free rate of 6%, $500 of capital required • No frictional COC, taxes or expenses • Calculation • Initial assets of $1500 grow to $1590, leaving $550 for a 10% return (COC) • Risk margin is $18.87 = 1000 - 1040/1.06

11. RM and COC Example, Cont. • Which comes first, the RM or the COC? • Each implies the other • In determining a fair premium, it must be the risk margin: • Products have different levels of risk • What COC should a riskless coverage have? • Thus, the COC is not fixed for an insurer -- it varies by product

12. Allocating Capital Costs • For pricing or performance measurement, must allocate capital costs to product • If we know the RM, then we need to allocate the frictional COC • If we don’t know the RM, and use a COC pricing model, then we allocate both frictional and base COC

13. Capital Allocation • In order to assess capital costs by product, we first need to allocate capital to product • There are many methods • Lots of ad hoc models • Very few economically sound models • One of them is the general Myers-Read method

14. Myers-Read Method • Uses the expected default (PH deficit) as a solvency measure • Others, such as default (ruin) probability will also work (and may be better) • Major assumptions • predetermined capital ratios exist: • A marginal change in the line mix keeps the default measure at a constant rate:

15. More on M-R Model • Other inputs • Probability distribution of loss and asset values • Means, correlations and volatilities • We solve for capital ratio • Result: • Beta is covariance/variance • Z is distribution-dependent

16. Loss Beta • Relevant risk measure for capital allocation is loss beta • Volatility, correlation with portfolio and weight determine loss beta • Strong parallel with asset pricing, CAPM, portfolio optimization • Capital allocation is exact; no overlap • No order dependency

17. Numerical Example

18. Application to Coverage Layers • For policy/treaty, capital allocation to layer depends on covariance of layer with that of unlimited loss • Layer covariance depends only on loss size distribution • Layer beta and capital/loss increase with limits

19. 25.00 20.00 15.00 10.00 5.00 0.00 0 50 100 150 200 250 300 350 400 x General Layer Beta Properties • Monotonic increasing with layer, with zero layer beta at lowest point layer • Generally unbounded Legend: RHS top to bottom Pareto Lognormal Exponential Gamma Normal

20. Practical Applications • Best measure of capital is economic (fair) value • As an approximation, capital is proportional to the loss/layer beta • For allocating a company’s capital, the relevant time horizon is one year • Allocation base is reserves plus next year’s AY incurred losses

21. Summary • Importance of frictional costs in theory of solvency and capital allocation • Myers-Read method is economically sound, with friendly (to user) results • We’ve still got a long road ahead before common agreement on capital allocation methodology

22. Further Reading • John Hancock, Paul Huber, Pablo Koch, 2001The economics of insurance: How insurers create value for shareholders, Swiss Re Publishinghttp://www.swissre.com/ • Myers, Read, 2001, Capital Allocation for Insurance Companies, Journal of Risk and Insurance, 68:4, 545-580 • Butsic, 1999, Capital Allocation for Property Liability Insurers: A Catastrophe Reinsurance Application. Casualty Actuarial Society Forum, Fall 1999http://www.casact.org/pubs/forum/99spforum/99spftoc.htm

23. Further Reading II • Butsic, Cummins, Derrig, Phillips, 2000, The Risk Premium Project, Phase I and II Report, CAS Website, http://casact.org/cotor/rppreport.pdf